What are the actual rights that are being defended or attacked? I have a few different formulations in mind:

On the side of the barber:
1) An individual has the right to offer or refuse service or exchange of goods with whomever they please in an agreement that does not violate established rights of the receiver.

On the side of the haircut:
1) An individual has a right to receive any service or any good they desire from any party who offers that service or good to any other party (w/out being contra established rights). As long as the individual receives the service from someone, their rights have been met.
2) An individual has a right to receive any service or any good they desire from any particular party who offers that service or good to any other party (usual caveat). The individual must receive the service/good from the party they desire, so long as that party also offers the service/good to others.

Whether the context is essential goods or services, the latter is harder to defend without denying that right of the barber. If the former collapses into the latter, which seems possible but not necessarily plausible in a sufficiently diverse market, then we're stuck in the same scenario. Since I take this to be where the conflict lies, the question, to me, is whether the barber should actually have the right to refuse service he offers to others, so long as it does not violate established rights.

The premise, I think, is that to refuse the barber this right would be refusing him essential autonomy and liberty. However, I take it for granted that we have the right to enforce upon individuals certain rules that are conducive to success in the market more broadly, in part because market activity, especially in a contemporary context, is not really that private. We don't let companies bait-and-switch, for example, because it undermines confidence in the market more generally. I think a similar argument can be extended to arbitrary refusal of service. It undermines confidence in the market and seems contrary to the very purpose of offering services and goods.

I'm not claiming that consumers are then permitted to do everything. Since every exchange involves two parties, consumers must also act in a way that does not undermine the confidence of sellers, and this includes, say, berating an employee, ignoring debts, etc.

What BMats has pointed towards is a challenge I'm somewhat sympathetic to but which requires a great deal of work. When you ask what ultimately grounds all rights, and when we start with individual liberty and autonomy as the fundamental principle, we start to run into chaos because individuals can have mutually exclusive views of their liberty and autonomy when interacting with each other. Out of that, I think there is a bit of a practical need to acknowledge that liberty and autonomy must be restricted in some sense, and that is when we start to ask about what sort of society you like. You then try to actualize the most practical rights to achieve that sort of society.

I think this is where I depart from libertarianism. I do not think that the best society can be actualized by libertarianism. I find socialist libertarianism extremely attractive, particularly on theoretical grounds, but I try not to be distracted by utopian ideas when the evidence on the ground seems to point in the other direction.

Yes, I do believe that politics, economics, and ethics are fundamentally pragmatic in nature.